Quantifying Predictive Uncertainty as a Function of Calibration Data Quantity and Model Resolution Using an Adjoint State Monte Carlo Technique

Model selection and calibration is a major concern in hydrology, as high-dimensional parameter spaces representing highly heterogeneous space-random fields are estimated from data in low-dimensional observation spaces. Due to extreme nonuniqueness, nonlinearity in parameters, and objective functions with multiple local minima, it is unclear how much physical information is imparted by the standard calibration strategy of reducing the residual error between observations and calibrated model predictions. Further, adding resolution to a model generally makes it increasingly easy to fit but (because of increasing non-uniqueness) decreases the amount of information it contains when calibrated. Thus, it is difficult to know how much predictive validity a calibrated hydrologic model exhibits, its optimal complexity, and also how much information is required to constrain it to a certain degree of accuracy.

We address this matter directly: we derive analytical adjoint equations to problems of interest and express their spatially-heterogeneous parameters as orthogonal function expansions. We then directly compute error envelopes for given calibration data quantities and model resolutions by Monte Carlo: selecting thousands of equiprobable random initial parameterizations and performing adjoint state steepest-descent local optimization for each to find a best match to observations. By recording quantities of interest (QoIs) for each initial parameterization and performing kernel density estimation, we arrive at empirical probability distributions for QoI uncertainty for a given model resolution, data quantity, and QoI fit. This direct approach is possible because adjoint state computational cost is determined by observable space dimension rather than parameter space dimension, as in traditional optimization. It is thus ideally suited for exploring highly underdetermined hydrologic inverse problems.

We demonstrate our method by quantifying expected error in a transmissivity field reconstructed from randomly located head measurements as a function of the number of said measurements for various amounts of boundary condition information. We also discuss the utility of our approach for model complexity selection and its superiority to techniques such as the so-called information criteria.